## Ambient spaces for cones generated by matricial inner products.(English)Zbl 1083.15034

This paper provides a short expository survey (without proofs) of the characterizations and properties of four ambient spaces for cones generated by matricial inner products on the space $${\mathcal M}_n$$ of complex $$n\times n$$ matrices. Results about the spaces of Hermitian-preserving transformations, real-preserving transformations, selfadjoint transformations, and the space of all transformations $$T$$ such that $$\langle T(A),A\rangle$$ is Hermitian for all matrices $$A\in {\mathcal M}_n$$ are examined.

### MSC:

 15B48 Positive matrices and their generalizations; cones of matrices 15B57 Hermitian, skew-Hermitian, and related matrices 15-02 Research exposition (monographs, survey articles) pertaining to linear algebra